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Number-theoretic Approaches to the Description of Crystal Structures

Hornfeck, Wolfgang (2012) Number-theoretic Approaches to the Description of Crystal Structures. 27th European Crystallographic Meeting (ECM27), 6.-11. Aug. 2012, Bergen, Norwegen.

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Abstract

Crystallographic group theory is commonly regarded as the method of choice for the description of crystal structures and related phenomena such as phase transformations, antiphase domain formation and twinning. Symmetry alone, however, merely comprises a qualitative picture. On the contrary, number theory seems to be far less explored within a crystallographic context, whereas it possibly allows for a more quantitative arithmetic approach. An illustrative example is given by so-called multiplicative congruential generators (MCGs), i.e. recurrence relations of the form Z(n+1) = m Z(n) (mod M), first introduced by Lehmer for the generation of pseudorandom number sequences. However, Marsaglia later noted that MCGs exhibit an intrinsic sublattice structure, which was shown to have some implications for the description of crystal structures. Another example is due to the bit-reversal, quasirandom number sequences of van der Corput, which exhibit features related to quasiperiodic binary substitution tilings. Both approaches share a common basis, as the integer sequences involved for the generation of the corresponding two- or three-dimensional point sets and their coordinate description are mere permutations of a finite set of successive natural numbers. Notably both methods have their primary application in the generation of random numbers, which sheds some light on the subtle interrelations between distinct states of order. Following the aforementioned number-theoretic construction principles allows for the systematic (algorithmic) generation of artificial crystal structures (permutation structures), as well as their combinatorial enumeration and classification. The peculiar structure of beta-Mn has already identified as a near-miss to these envisioned permutation structures.

Document Type:Conference or Workshop Item (Poster)
Title:Number-theoretic Approaches to the Description of Crystal Structures
Authors:
AuthorsInstitution or Email of Authors
Hornfeck, Wolfgangwolfgang.hornfeck@dlr.de
Date:9 August 2012
Status:Accepted
Keywords:Permutations, Random Numbers, Quasicrystals, Intermetallic Compounds
Event Title:27th European Crystallographic Meeting (ECM27)
Event Location:Bergen, Norwegen
Event Type:international Conference
Event Dates:6.-11. Aug. 2012
Organizer:European Crystallographic Association
HGF - Research field:Aeronautics, Space and Transport (old)
HGF - Program:Space (old)
HGF - Program Themes:W FR - Forschung unter Weltraumbedingungen (old)
DLR - Research area:Space
DLR - Program:W FR - Forschung unter Weltraumbedingungen
DLR - Research theme (Project):W - Vorhaben Materialwissenschaftliche Forschung (old)
Location: Köln-Porz
Institutes and Institutions:Institute of Materials Physics in Space
Deposited By: Dr. Wolfgang Hornfeck
Deposited On:16 Aug 2012 09:28
Last Modified:16 Aug 2012 09:28

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